Stable and Unstable Subspaces under Diffeomorhpism (Hyperbolic dynamics)

Question

I'm trying to wrap my head around hyperbolic dynamics without much knowledge of differentiable manifolds, so this may be trivial.

In Intro. to Dynamical Systems (Brin & Stuck), an hyperbolic set is defined as follows. Given a $C^1$ Riemannian manifold M and a diffeomorphism $f$, a compact $f$-invariant subset $\Lambda$ is called hyperbolic if there are $\lambda \in (0,1)$, $C>0$ such that for $x \in \Lambda$, there are subspaces $E^s(x) \subset T_x M$ and $E^u(x) \subset T_x M$ satisfying:

1. $T_x M = E^s(x) \oplus E^u(x)$
2. $||df_x^n v|| \leq C\lambda^n ||v||$ for $v \in E^s(x)$
3. $||df_x^{-n} v|| \leq C\lambda^n ||v||$ for $v \in E^u(x)$
4. $df_xE^s(x) = E^s(f(x))$ and $df_xE^u(x) = E^u(f(x))$

Now, if all four are satisfied for some $x \in M$, are they then also satisfied for $f(x)$? And how does one go about proving the fourth condition?

I'm especially motivated by the following question (in preparation for an exam, not a homework): for a hyperbolic fixed point $p$ of $f$ and $q$ a transverse homoclinic point of $p$, proof that the union of orbit of $p$ with $q$ is a hyperbolic set.

It is easy to show that the union is closed and $f$-invariant and that $q$ satisfies the first three conditions. But I have no clue how to proof the fourth or how to extend this to the rest of the orbit of $q$.

Answer 1

For the example that you describe, only properties $2$ and $3$ are not immediate. Note that a priori the transverse homoclinic point has no hyperbolicity! But under iteration $f^n(q)$ is close to $p$. Since $f$ is $C^1$ you get properties $2$ and $3$ (for example this is what makes the $\lambda$-lemma work).

Property $4$ is immediate since the stable and unstable spaces can only be the tangent spaces, what else since we know that stable and unstable spaces are defined uniquely (and since the tangent spaces to the stable and unstable manifolds of $p$ satisfy automatically property $4$)?

On your first question, whether it also holds for $f(x)$. The hypothesis is for all $x\in\Lambda$ anyways, but yes you can induce stable and unstable spaces at $f^2(x)$ and prove all properties provided that you allow for a larger $C$.